You seem to be using $X$ for both the measure space and limit of summation (I don't have the reputation to edit).
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Noah SteinOct 20 '10 at 10:40

I don't think it's right that the answer is yes for an irrational rotation and the function $f$ is taken to be $L^1$. There is a theorem called the transference principle (I think of it as a photocopying machine) allowing you to transfer a counterexample that diverges for one measure-preserving transformation to any other aperiodic measure-preserving transformation.
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Anthony QuasJul 19 '14 at 3:45

2 Answers
2

Here is a partial answer: Mate Wierdl proved that the limit exists almost everywhere if $f \in L^r (\mu)$ for some $r>1$. See "Pointwise Ergodic Theorem along the Prime Numbers".

Also, there is a recent article by Trevor Wooley and Tamar Ziegler ("Multiple Recurrence and Convergence along the Primes") which proves $L^2$ convergence and multiple recurrence for more complicated ergodic averages of this kind.

Patrick LaVictoire shows that the answer is negative if you ask for all of $L^1$.

His paper is Universally L^1-Bad Arithmetic Sequences (to appear in Journal d'Analyse Mathematique). The paper extends results of Buczolich and Mauldin (who showed a negative result if you sum along the squares). The paper can be found online at http://arxiv.org/abs/0905.3865